We provide new results and new proofs of results about the torsion of curves in $${\mathbb{R}^3}$$R3. Let $${\gamma}$$γ be a smooth curve in $${\mathbb{R}^3}$$R3 that is the graph over a simple closed curve in $${\mathbb{R}^2}$$R2 with positive curvature. We give a new proof that if $${\gamma}$$γ has nonnegative (or nonpositive) torsion, then $${\gamma}$$γ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $${\mathbb{R}^{2,1}}$$R2,1 which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.